Here we present \(\mathrm {L_{p}^\ast }\) , a variant of \(\mathrm {WL}^\ast \)  [1]}. Similarly, it uses an observation table \(O_{Pre,Suf}\)  for storing outcomes of \(\mathbf {MQ}_{\mathrm {L_{p}^\ast }}\) , where \(Pre\subset \Sigma ^\ast \)  is the set of prefixes (stored in row indices) and \(Suf\subset \lbrace \$\rbrace \cup \Sigma ^{+}\)  is the set of suffixes (stored in column indices).
For every
\(p\in Pre\)  and \(s\in Suf\) , \(O_{Pre,Suf}[p,s] = \mathbf {MQ}_{\mathrm {L_{p}^\ast }}(p s) = \pi ^\ell _{q_{\mathrm {in}}}(p s)\) , where \(\pi ^\ell \)  is defined in (REF ).
Unlike \(\mathrm {WL}^\ast \) , \(Pre\)  is divided in two parts [2]}, namely RED which are the rows used to construct states, and BLUE which are the rows representing continuations. The algorithm ensures that for every row in RED all its continuations are in \(Pre\) .
\(\mathrm {L_{p}^\ast }\)  consists of three main steps. The first one expands \(O_{Pre,Suf}\)  through the use of \(\mathbf {MQ}_{\mathrm {L_{p}^\ast }}\)  until it becomes closed and consistent. The second one constructs a hypothesis automaton using a greedy clustering technique rather than DBScan. The third one calls \(\mathbf {EQ}_{\mathrm {L_{p}^\ast }}\)  with the proposed hypothesis. For this we adapted Hopcroft-Karp algorithm for checking \(t\) -similarity. When \(A\)  and \(\widehat{A}\)  are found to be not \(t\) -similar, \(\mathbf {EQ}_{\mathrm {L_{p}^\ast }}\)  returns a counterexample which is added to RED together with all its prefixes, and all their continuations to BLUE (provided they are not already in RED). These steps are repeated as long as \(\mathbf {EQ}_{\mathrm {L_{p}^\ast }}\)  yields a counterexample, otherwise it stops and returns the last hypothesis.
